On Volumes of Permutation Polytopes

Katherine, Burggraf,; Loera, Jesús A. De; Omar, Mohamed

doi:10.1007/978-3-319-00200-2_5

Cited by 7 publications

(4 citation statements)

References 32 publications

(56 reference statements)

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“…In [45], Costa et al presented a formula counting the number of faces tridiagonal Birkhoff polytopes. Volumes of permutation polytopes were studied in [32].…”

Section: Further Research Directions and More Open Problemsmentioning

confidence: 99%

Combinatorics and geometry of transportation polytopes: An update

Loera¹,

Kim²

2014

Contemporary Mathematics

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Abstract. A transportation polytope consists of all multidimensional arrays or tables of non-negative real numbers that satisfy certain sum conditions on subsets of the entries. They arise naturally in optimization and statistics, and also have interest for discrete mathematics because permutation matrices, latin squares, and magic squares appear naturally as lattice points of these polytopes.In this paper we survey advances on the understanding of the combinatorics and geometry of these polyhedra and include some recent unpublished results on the diameter of graphs of these polytopes. In particular, this is a thirty-year update on the status of a list of open questions last visited in the 1984 book by Yemelichev, Kovalev and Kravtsov and the 1986 survey paper of Vlach.

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“…In [45], Costa et al presented a formula counting the number of faces tridiagonal Birkhoff polytopes. Volumes of permutation polytopes were studied in [32].…”

Section: Further Research Directions and More Open Problemsmentioning

confidence: 99%

Combinatorics and geometry of transportation polytopes: An update

Loera¹,

Kim²

2014

Contemporary Mathematics

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“…Studying variations of the Birkhoff polytope is not uncommon. For example, permutation polytopes, subpolytopes of B n whose vertices form a subgroup of S n , have been studied by, for example, Burggraf, De Loera, and Omar [11], who studied their volumes, and Onn [24], who studied their low-dimensional skeletons and combinatorial types. Another important variation is the class of transportation polytopes, in which row and column sums may be numbers other than 1, and two rows or columns do not necessarily need to sum to the same value.…”

Section: The Birkhoff Polytopementioning

confidence: 99%

Pattern-avoiding polytopes

Davis

Sagan

2018

European Journal of Combinatorics

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Two well-known polytopes whose vertices are indexed by permutations in the symmetric group S n are the permutohedron P n and the Birkhoff polytope B n . We consider polytopes P n (Π) and B n (Π), whose vertices correspond to the permutations in S n avoiding a set of patterns Π. For various choices of Π, we explore the Ehrhart polynomials and h * -vectors of these polytopes as well as other aspects of their combinatorial structure.For P n (Π), we consider all subsets Π ⊆ S 3 and are able to provide results in most cases. To illustrate, P n (123, 132) is a Pitman-Stanley polytope, the number of interior lattice points in P n (132, 312) is a derangement number, and the normalized volume of P n (123, 231, 312) is the number of trees on n vertices.The polytopes B n (Π) seem much more difficult to analyze, so we focus on four particular choices of Π. First we show that the B n (231, 321) is exactly the Chan-Robbins-Yuen polytope. Next we prove that for any Π containing {123, 312} we have h * (B n (Π)) = 1. Finally, we study B n (132, 312) and B n (123), where the tilde indicates that we choose vertices corresponding to alternating permutations avoiding the pattern 123. In both cases we use order complexes of posets and techniques from toric algebra to construct regular, unimodular triangulations of the polytopes. The posets involved turn out to be isomorphic to the lattices of Young diagrams contained in a certain shape, and this permits us to give an exact expression for the normalized volumes of the corresponding polytopes via the hook formula. Finally, Stanley's theory of (P, ω)-partitions allows us to show that their h * -vectors are symmetric and unimodal.Various questions and conjectures are presented throughout.

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“…Besides [LM94] and [SL15] (and the follow-on [SL16]), there have been a few papers on analytic formulae for volumes of polytopes that naturally arise in mathematical optimization; see [KLS97], [Ste94], [BDLO13], [ABD10], [Sta86]. But none of these works has attempted to apply their ideas to the low-dimensional polytopes that naturally arise in sBB, or even to apply their ideas to compare relaxations.…”

Section: Literature Reviewmentioning

confidence: 99%

Experimental Validation of Volume-Based Comparison for Double-McCormick Relaxations

Speakman

Lee

2017

Integration of AI and OR Techniques in Constraint Programming

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Volume is a natural geometric measure for comparing polyhedral relaxations of non-convex sets. Speakman and Lee gave volume formulae for comparing relaxations of trilinear monomials, quantifying the strength of various natural relaxations. Their work was motivated by the spatial branch-and-bound algorithm for factorable mathematical-programming formulations. They mathematically analyzed an important choice that needs to be made whenever three or more terms are multiplied in a formulation. We experimentally substantiate the relevance of their main results to the practice of global optimization, by applying it to difficult box cubic problems (boxcup). In doing so, we find that, using their volume formulae, we can accurately predict the quality of a relaxation for boxcups based on the (box) parameters defining the feasible region. * Supported in part by ONR grant N00014-14-1-0315.

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On Volumes of Permutation Polytopes

Cited by 7 publications

References 32 publications

Combinatorics and geometry of transportation polytopes: An update

Combinatorics and geometry of transportation polytopes: An update

Pattern-avoiding polytopes

Experimental Validation of Volume-Based Comparison for Double-McCormick Relaxations

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